Poincaré sections for the horocycle flow in covers of $$\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})$$ and applications to Farey fraction statistics

2016 ◽  
Vol 179 (3) ◽  
pp. 389-420 ◽  
Author(s):  
Byron Heersink
Keyword(s):  
Horocycle Flow ◽  
Poincaré Sections ◽  
Farey Fraction ◽  
Poincare Sections

Hamiltonian Chaos in a Model Alveolus

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
F. S. Henry ◽  
F. E. Laine-Pearson ◽  
A. Tsuda
Keyword(s):  
Reynolds Number ◽  
Fixed Point ◽  
Lyapunov Exponent ◽  
Fine Particles ◽  
Hamiltonian Dynamics ◽  
Maximal Lyapunov Exponent ◽  
Poincaré Sections ◽  
Pulmonary Acinus ◽  
Hamiltonian Chaos ◽  
Poincare Sections

In the pulmonary acinus, the airflow Reynolds number is usually much less than unity and hence the flow might be expected to be reversible. However, this does not appear to be the case as a significant portion of the fine particles that reach the acinus remains there after exhalation. We believe that this irreversibility is at large a result of chaotic mixing in the alveoli of the acinar airways. To test this hypothesis, we solved numerically the equations for incompressible, pulsatile, flow in a rigid alveolated duct and tracked numerous fluid particles over many breathing cycles. The resulting Poincaré sections exhibit chains of islands on which particles travel. In the region between these chains of islands, some particles move chaotically. The presence of chaos is supported by the results of an estimate of the maximal Lyapunov exponent. It is shown that the streamfunction equation for this flow may be written in the form of a Hamiltonian system and that an expansion of this equation captures all the essential features of the Poincaré sections. Elements of Kolmogorov–Arnol’d–Moser theory, the Poincaré–Birkhoff fixed-point theorem, and associated Hamiltonian dynamics theory are then employed to confirm the existence of chaos in the flow in a rigid alveolated duct.

Perturbing variables in poincaré sections to suppress chaos

2008 ◽  
pp. 339-340
Author(s):  
J. M. Gutiérrez ◽  
A. Iglesias ◽  
J. Güémez ◽  
M. A. Matías
Keyword(s):  
Poincaré Sections ◽  
Poincare Sections

Simultaneous Resonances of Suspended Cables Subjected to Primary and Super-Harmonic Excitations in Thermal Environments

2019 ◽  
Vol 19 (12) ◽  
pp. 1950155
Author(s):  
Yaobing Zhao ◽  
Henghui Lin ◽  
Lincong Chen ◽  
Chenfei Wang
Keyword(s):  
Perturbation Method ◽  
Multiple Scales ◽  
Effect Of Temperature ◽  
Response Curves ◽  
Temperature Changes ◽  
Poincaré Sections ◽  
Thermal Environments ◽  
Direct Integration Method ◽  
Harmonic Excitations ◽  
Poincare Sections

This paper concerns with a suspended cable in thermal environments under bi-frequency harmonic excitations, with a focus placed on the effect of temperature changes on one type of simultaneous resonance. First, the nonlinear equation of motion in thermal environments is obtained for the in-plane displacement of the cable. Then, the Galerkin method is employed to reduce the partial differential equation to an ordinary one. Second, based on the discretized form of the governing equation, the method of multiple scales is employed to obtain the second-order approximate solutions, with the stability characteristics determined. Third, numerical results are presented by using the perturbation method, together with numerical integration by the following means: frequency-response curves, time-displacement curves, phase-plane diagrams, and Poincare sections. The direct integration method is utilized to verify the results obtained by the perturbation method, while revealing more nonlinear dynamic behaviors induced by temperature changes. Both the softening and/or hardening behaviors, and the switching between them are observed for the cable in thermal environments. The response amplitude of the cable is very sensitive to temperature changes, but the number of circles in the phase diagrams and the number of cluster points in Poincaré sections is independent of the thermal effects in most cases. Finally, the vibration characteristics of the cable for different thermal expansion coefficients and temperature-dependent Young’s moduli are also investigated.

scholarly journals Nonlinear dynamics and Poincaré sections to model gait impairments in different stages of Parkinson’s disease

2020 ◽  
Vol 100 (4) ◽  
pp. 3253-3276
Author(s):  
P. A. Pérez-Toro ◽  
J. C. Vásquez-Correa ◽  
T. Arias-Vergara ◽  
E. Nöth ◽  
J. R. Orozco-Arroyave
Keyword(s):  
Nonlinear Dynamics ◽  
Parkinson’S Disease ◽  
Parkinson's Disease ◽  
Poincaré Sections ◽  
Poincare Sections ◽  
Gait Impairments

Chaotic Motion of a Parametrically Excited Dielectric Elastomer

2020 ◽  
Vol 12 (03) ◽  
pp. 2050033
Author(s):  
Hamidreza Heidari ◽  
Amin Alibakhshi ◽  
Habib Ramezannejad Azarboni
Keyword(s):  
Time Integration ◽  
Nonlinear Behavior ◽  
Strain Energy Function ◽  
Nonlinear Ordinary Differential Equation ◽  
Dielectric Elastomer ◽  
Dissipation Function ◽  
Bifurcation Diagrams ◽  
Chaotic Motions ◽  
Poincaré Sections ◽  
Poincare Sections

In this paper, an effort is made to study the chaotic motions of a dielectric elastomer (DE). The DE is activated by a time-dependent voltage (AC voltage), which is superimposed on a DC voltage. The Gent strain energy function is employed to model the nonlinear behavior of the elastomeric matter. The nonlinear ordinary differential equation (ODE) in terms of the stretch of the elastomer governing the motion of the system is deduced using the Euler–Lagrange method and the Rayleigh dissipation function. This ODE is solved via the use of a time integration-based solver. The bifurcation diagrams of Poincaré sections are generated to identify the chaotic domains. The largest Lyapunov exponents (LLEs) are illustrated for validation of the results obtained by the bifurcation diagrams. Various types of motion for the system are precisely discussed through the depiction of stretch-time responses, phase-plane diagrams, Poincaré sections and power spectral density (PSD) diagrams. The results reveal that the damping coefficient plays an influential role in suppressing the chaos phenomenon. Besides, the initial stretch of the elastomer could affect the chaotic interval of system parameters.

scholarly journals Poincaré sections of Hamiltonian systems

1996 ◽  
Vol 95 (2-3) ◽  
pp. 171-189 ◽  
Author(s):  
E.S. Cheb-Terrab ◽  
H.P. de Oliveira
Keyword(s):  
Hamiltonian Systems ◽  
Poincaré Sections ◽  
Poincare Sections

Global Dynamics of an Autoparametric System With Multiple Pendulums

2005 ◽  
Vol 1 (1) ◽  
pp. 35-46 ◽  
Author(s):  
Ashwin Vyas ◽  
Anil K. Bajaj
Keyword(s):  
Initial Conditions ◽  
Hamiltonian Formulation ◽  
Forced Response ◽  
Global Dynamics ◽  
Equilibrium Solutions ◽  
External Excitation ◽  
Poincaré Sections ◽  
Poincare Sections ◽  
Hamilton's Equations ◽  
Hamilton’S Equations

The Hamiltonian dynamics of a resonantly excited linear spring-mass-damper system coupled to an array of pendulums is investigated in this study under 1:1:1:…:2 internal resonance between the pendulums and the linear oscillator. To study the small-amplitude global dynamics, a Hamiltonian formulation is introduced using generalized coordinates and momenta, and action-angle coordinates. The Hamilton’s equations are averaged to obtain equations for the first-order approximations to free and forced response of the system. Equilibrium solutions of the averaged Hamilton’s equations in action-angle or comoving variables are determined and studied for their stability characteristics. The system with one pendulum is known to be integrable in the absence of damping and external excitation. Exciting the system with even a small harmonic forcing near a saddle point leads to stochastic response, as clearly demonstrated by the Poincaré sections of motion. Poincaré sections are also computed for motions started with initial conditions near center-center, center-saddle and saddle-saddle-type equilibria for systems with two, three and four pendulums. In case of the system with more than one pendulum, even the free undamped dynamics exhibits irregular exchange of energy between the pendulums and the block. The increase in complexity is also demonstrated as the number of pendulums is increased, and when external excitation is present.

Band structure and quantum Poincaré sections of a classically chaotic quantum rippled channel

1996 ◽  
Vol 53 (4) ◽  
pp. 3271-3283 ◽  
Author(s):  
G. A. Luna-Acosta ◽  
Kyungsun Na ◽  
L. E. Reichl ◽  
A. Krokhin
Keyword(s):  
Band Structure ◽  
Poincaré Sections ◽  
Poincare Sections

scholarly journals ORDER AND CHAOS IN ROTO-VIBRATIONAL STATES OF ATOMIC NUCLEI

1995 ◽  
Vol 04 (03) ◽  
pp. 625-636 ◽  
Author(s):  
V.R. MANFREDI ◽  
L. SALASNICH
Keyword(s):  
Numerical Results ◽  
Numerical Calculations ◽  
Vibrational States ◽  
Deformed Nuclei ◽  
Poincaré Sections ◽  
Atomic Nuclei ◽  
Analytical Criterion ◽  
Spectral Statistics ◽  
Poincare Sections

Using a classical analytical criterion (that of curvature) and numerical results (Poincare sections and spectral statistics), a transition order-chaos-order in the roto-vibrational model of atomic nuclei has been shown. Numerical calculations were performed for some deformed nuclei.

Bifurcations and Chaos in Voice Signals

1993 ◽  
Vol 46 (7) ◽  
pp. 399-413 ◽  
Author(s):  
Hanspeter Herzel
Keyword(s):  
Early Diagnosis ◽  
Speech Production ◽  
Deterministic Chaos ◽  
Brain Disorder ◽  
Poincaré Sections ◽  
Physical Mechanisms ◽  
Rich Variety ◽  
Low Dimensional ◽  
Poincare Sections

The basic physical mechanisms of speech production is described. A rich variety of bifurcations and episodes of irregular behaviour are observed. Poincare´ sections and the analysis of the underlying attractor suggest that these noise-like episodes are low-dimensional deterministic chaos. Possible implications for the very early diagnosis of brain disorder are discussed.

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